Finite Group Databasebeta

← #33137

Subgroups — #33137

Order 256.

426 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic
2C2×#405C211yesmaximal, characteristic, complemented
3#2625C211yesmaximal, characteristic, complemented
4#2624C211yesmaximal, characteristic
5#1968C212yesmaximal, complemented
6#1941C212yesmaximal, complemented
7#1936C212yesmaximal, complemented
8#1935C212yesmaximal, complemented
9#1895C211yesmaximal, characteristic
10#1894C211yesmaximal, characteristic, complemented
11#1807C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×#117C2211yescharacteristic
18#465—24no—
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2211yescharacteristic
23C2×#102C2212yes—
24C2×#102C2211yescharacteristic
25#405—24no—
26C2×#97C2212yes—
27C2×C4×C8C2212yes—
28#400—21no—
29#399—21no—
30#398—21no—
31#397—21no—
32#397—22no—
33#394—22no—
34#391C2212yes—
35#390C2212yes—
36#389C2212yes—
37#389C2212yes—
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C2×SD16—22no—
48C2×SD16—24no—
49C2×M16—22no—
50C2×M16—24no—
51C22×C8—24no—
52C22×C8D414yes—
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58#121—22no—
59#121—22no—
60#117—22no—
61#117—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D412yes—
78C2×C42C2312yes—
79#106—22no—
80#106—24no—
81#103—22no—
82#103—24no—
83#102—22no—
84#102—24no—
85#97—24no—
86C4×C8—24no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93C24—21no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×D4—21no—
100C2×D4—22no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—42no—
104C2×D4—82no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4#4112yes—
117C22×C4C2411yescharacteristic
118SD16—44no—
119M16—28no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—28no—
124C2×C8C2×D412yes—
125C2×C8C2×D412yes—
126C2×C8#4114yes—
127C2×C8#4114yes—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—82no—
141C22⋊C4—24no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C22⋊C4—81no—
147C22⋊C4—81no—
148C42—22no—
149C42—22no—
150C23—22no—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—41no—
155C23—42no—
156C23—41no—
157C23—42no—
158C23—41no—
159C23#12112yes—
160C23C22×D411yescharacteristic
161Q8—41no—
162Q8—41no—
163Q8—42no—
164D4—41no—
165D4—41no—
166D4—42no—
167D4—84no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4#12112yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4112yes—
194C2×C4C2×#4112yes—
195C2×C4C2×#4111yescharacteristic
196C8—22no—
197C8—24no—
198C8#12314yes—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22—82no—
206C22—82no—
207C22—81no—
208C22—81no—
209C22C2×#12111yescharacteristic
210C22#53212yes—
211C22#57711yescharacteristic
212C22#57712yes—
213C22#57711yescharacteristic
214C4—22no—
215C4—42no—
216C4—42no—
217C4—81no—
218C4—82no—
219C4—82no—
220C4—81no—
221C4—81no—
222C4C2×#12311yescharacteristic
223C4#52511yescharacteristic
224C4#52912yes—
225C2—42no—
226C2—161no—
227C2#285411yescharacteristic
228C2#305711yescharacteristic
229C2#306011yescharacteristic
230C2#317312yes—
231C2#317412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#405C2semidirect
3. #2625C2semidirect
5. #1968C2semidirect
6. #1941C2semidirect
7. #1936C2semidirect
8. #1935C2semidirect
10. #1894C2semidirect
11. #1807C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#4652
C2×#4052
#26252
#26242
#19682
#19412
#19362
#19352
#18952
#18942
#18072